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Roger Torres, Jean-Charles de Hemptinne, I. Machin
To cite this version:
Improving the Modeling of Hydrogen Solubility
in Heavy Oil Cuts Using an Augmented
Grayson Streed (AGS) Approach
R. Torres
1*, J.-C. de Hemptinne
2and I. Machin
11 PDVSA Intevep, Urb. Santa Rosa, Sector El Tambor, Los Teques 1201 - Venezuela 2 IFP Energies nouvelles, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - France e-mail: [email protected] – [email protected] – [email protected]
* Corresponding author
Résumé — Modélisation améliorée de la solubilité de l’hydrogène dans des coupes lourdes par l’approche de Grayson Streed Augmenté (GSA) — La méthode de Grayson Streed (GS) [Grayson
H.G. and Streed C.W. (1963) 6th World Petroleum Congress, Frankfurt am Main, Germany, 19-26 June, pp. 169-181] est souvent préconisée dans l’industrie pour calculer la solubilité de l’hydrogène dans des coupes pétrolières. Il se fait cependant que sa précision se dégrade rapidement pour les coupes lourdes. Une amélioration est proposée dans ce travail, basée sur l’ajout d’un terme de Flory dans le calcul du coefficient d’activité.
L’étude de la solubilité de l’hydrogène dans les n-alcanes du n-C7 au n-C36fait apparaître que la constante de Henry diminue avec la masse molaire. L’analyse de ce comportement suggère la présence d’une déviation entropique à l’idéalité non prise en compte dans le modèle des solutions régulières. L’utilisation d’une correction de Flory permet de garder l’aspect prédictif du modèle. Elle nécessite néanmoins un nouveau calage de certains paramètres de la corrélation d’origine pour l’hydrogène. Le modèle qui résulte se comporte mieux pour les composés lourds et aromatiques.
La qualité du nouveau modèle de Grayson Streed Augmenté (GSA) est évaluée sur des données de solubilité d’hydrogène dans des coupes pétrolières issues de Cai et al. [Cai H.Y. et al. (2001) Fuel 80, 1055-1063] ainsi que Lin et al. [Lin H.M. et al. (1981) Ind. Eng. Chem. Process Des. Dev. 20, 2, 253-256]. L’importance de la caractérisation de ces coupes est mise en avant. Une analyse de sensibilité montre qu’une perturbation du paramètre de solubilité a un effet beaucoup plus important que pour les autres paramètres. Il en résulte qu’un grand soin doit être apporté au calcul de cette grandeur. La prédiction de la solubilité de l’hydrogène dans des fractions pétrolières lourdes et dans des charbons liquéfiés a été améliorée par rapport au modèle de Grayson Streed : une déviation absolue moyenne de 30 % est obtenue pour GSA, à comparer avec 55 % avec la méthode GS, avec les données utilisées dans un domaine de 80-380 °C et 6,3-258,9 bar.
Abstract — Improving the Modeling of Hydrogen Solubility in Heavy Oil Cuts Using an Augmented Grayson Streed (AGS) Approach — The Grayson Streed (GS) method [Grayson H.G. and Streed C.W.
(1963) 6th World Petroleum Congress, Frankfurt am Main, Germany, 19-26 June, pp. 169-181] is often used by the industry for calculating hydrogen solubility in petroleum fluids. However, its accuracy becomes very bad when very heavy fluids are considered. An improvement is proposed in this work, based on a Flory-augmented activity coefficient model.
Hydrogen solubilities in n-alkanes from n-C7up to n-C36have been investigated and a decreasing Henry
constant with molecular weight is evidenced. The analysis of the Henry constant behaviour with Copyright © 2013, IFP Energies nouvelles
DOI: 10.2516/ogst/2012061
InMoTher 2012 - Industrial Use of Molecular Thermodynamics InMoTher 2012 - Application industrielle de la thermodynamique moléculaire
LIST OF SYMBOLS
A Derived parameter of the Redlich-Kwong equation
AAD Average absolute deviation
B Derived parameter of the Redlich-Kwong equation
D Relative deviation
Δhivap Molar heat of vaporization of component i
Δhbvap Molar heat of vaporization at normal boiling point
Δuivap Molar energy of vaporization of component i
fiL Fugacity of component i in the liquid phase
fiL* Fugacity of pure liquid i at T and P of the mixture
fiV Fugacity of component i in the gas phase
GE Excess Gibbs energy
HE Excess enthalpy
Hi Henry constant for component i (solute)
Ki Vaporization equilibrium ratio of component i
kij Binary interaction coefficient
N Total number of data points
P Pressure
Pci Critical pressure of component i
Pr Reduced pressure
Psσ Vapour pressure of the solvent
R Universal gas constant
SE Excess entropy
SG Specific gravity at 15.5°C
T Temperature
Tb Normal boiling temperature
Tci Critical temperature of component i
Tr Reduced temperature
vi Molar volume of component i
xi Mole fraction of component i in the liquid phase
yi Mole fraction of component i in the vapour phase
z Compressibility factor
Greek Letters
γi Activity coefficient of component i at T of the mixture
γi∞ Activity coefficient of component i at infinite dilution
δi Solubility parameter of component i
δ –
Solubility parameter for the solution
ρ Density
ρT Density at given temperature T
φi Volume fraction of component i in liquid solution
ϕiL* Fugacity coefficient of pure liquid i at system condition
ϕ(0) Fugacity coefficient of “simple fluids” in liquid state ϕ(1) Fugacity coefficient correction factor
ϕiV Fugacity coefficient of component i in vapour mixture
ω Acentric factor
INTRODUCTION
As resources of light and conventional crude oils in the world are being depleted, an ever increasing use is made of extra heavy oil and tar sands, which must be converted into clean liquid fuels. The upgrading is achieved through processes such as hydrotreating or hydrocracking. Design and opera-tion of equipments for such processes require the knowledge of the hydrogen solubility in increasingly heavy petroleum mixtures. Reliable estimates of the amount of hydrogen in the hydrocarbon and oil fraction are therefore necessary.
In this work, it is proposed to use the Grayson Streed method to predict hydrogen solubility. The advantage of this method is that it is predictive, as it requires no interaction parameter data. However, for heavy hydrocarbons (greater than C15) the predicted hydrogen solubility may deviate quite bit from experimental values (de Hemptinne et al., 2012). Therefore, it is proposed to add a correction as improvement to this method.
1 LITERATURE REVIEW
Prediction of gas solubility in a liquid solvent is based on the general principles of Vapour-Liquid Equilibrium (VLE) which is formulated through equality of fugacities for each compound between the two phases (Riazi, 2005):
fiV= fiL (1)
molecular weight suggests a simple improvement to the model, using a Flory entropic contribution, thus keeping its predictive character. This improvement led to the necessity of refitting a number of fundamental hydrogen parameters. The resulting model behaves better for heavy components and for aromatics.
The petroleum fractions evaluated with the Augmented Grayson-Streed (AGS) model are taken from Cai
et al. [Cai H.Y. et al. (2001) Fuel 80, 1055-1063] and Lin et al. [Lin H.M. et al. (1981) Ind. Eng. Chem. Process Des. Dev. 20, 2, 253-256]. The importance of the petroleum fluid characterization is stressed. A
for all i components at constant T and P, where fiVand fiLare
the fugacities of component i in the gas and liquid phase. These fugacities may be calculated through fugacity coeffi-cients ϕiV and ϕiL which leads to the following relation
(Riazi, 2005):
yiϕiV= xiϕiL (2)
where xiis the mole fraction of component i in the liquid
phase and yiis the mole fraction of component i in the vapour
phase. The fugacity of component i in the liquid phase fiLmay
also be calculated using the deviation from an ideal solution,
i.e. through activity coefficient, Equation (2) becomes (Riazi,
2005):
yiϕiVP = xiγifiL* (3)
where fiL*is the fugacity of pure liquid i at T and P of the
mixture. The activity coefficient γidepends both on
tempera-ture and composition xi.
The two forms are equivalent but the main difference between Equations (2) and (3) for VLE calculation is in their applications. Equation (2) is particularly useful when both ϕiVand ϕiL are calculated from an Equation of State (EoS), in
this case, we speak of a homogeneous method. Cubic EoS generally work well for VLE calculation of petroleum fluids at high pressures using Equation (2). This approach is generally preferable, as all thermodynamic relations will be naturally satisfied. This is also the only possible approach for calculations close to a critical point: the only way to satisfy the critical criterion that all phase properties tend to coincide is to use an identical model for all phases (de Hemptinne et al., 2012).
Very often, cubic equations of state are used for calculating ϕiVand ϕiL. As a result of the very highly supercritical nature
of hydrogen, the EoS must be tuned. Either a modified tem-perature dependence of the attraction parameter (e.g. Twu et
al., 1996) or an adequate binary interaction coefficient, that
depends on both temperature (e.g. Moysan et al., 1983) and the nature of the solvent must then be used. However, the parameters of these models have been adjusted in a restricted temperature range and for a limited number of hydrocarbons for which experimental data were available. As a result, an extra-polation of these interaction coefficients for higher temperatures such as those encountered in hydrotreatment conditions may be questionable (Ferrando and Ungerer, 2007).
More recently, molecular models, such as SAFT (Statistical Association Fluid Theory), have been used with success on this problem (Florusse et al., 2003; Le Thi et al., 2006; Tran
et al., 2009). In a similar way as what is conventionally done
in molecular simulation techniques (Ferrando and Ungerer, 2007), a group-contribution type approach is proposed. However, these molecular tools, although very promising, require a molecular description of the fluids, which is generally hard to realize for heavy petroleum fractions.
In the case of Equation (3), a different model is used to describe the behaviour of the liquid phase and another to model the vapour phase; in this case, we speak of a heteroge-neous approach. This is very useful when the liquid phase non-idealities are significant (de Hemptinne et al., 2012).
The choice of a method for calculating the solubility of hydrogen in heavy oil cuts is guided by two conflicting require-ments: the need for accuracy and a search for simplicity.
Shaw (1987) proposed such a correlation adapted to heavy coal liquids and bitumen. The correlation by Shaw provided good predictive results but has to our knowledge not been extensively used in the petroleum refining industry. Instead, the Grayson Streed model is often considered as a “reference” model to simulate hydrogen/hydrocarbon equilibrium. However, it performs poorly for the mixtures involving heavy hydro-carbons (Ferrando and Ungerer, 2007). Hence this model is not the most adequate to predict phase equilibrium data in the operating conditions of hydrotreatments of heavy cuts.
The method of Grayson Streed, available in most process simulators, is a good compromise between accuracy and sim-plicity. A quick review of the foundations of this method shows that improvements can be introduced for increasing accuracy without losing its simplicity. In this sense, our goal is to propose some variants that can be readily substituted to the original correlation. In the following section a brief review of the fundamentals of the Grayson Streed (Grayson and Streed, 1963) model is presented.
2 THE GRAYSON STREED MODEL (GS) 2.1 How It Works
The Grayson Streed method is based on a heterogeneous, asymmetric approach, in which the distribution coefficient Ki
is calculated as follows:
(4) where the three factors are calculated using a different model: the pure liquid fugacity coefficient, ϕiL*, is calculated using a
specific, corresponding states method. The liquid activity coefficient, γi, is calculated using the regular solution model
and the vapour phase fugacity coefficient, ϕiV, is computed
from the Redlich Kwong cubic equation of state. 2.1.1 The Pure Liquid Fugacity Coefficient
The fugacity coefficient of pure liquid is calculated with a Curl and Pitzer corresponding state correlation:
logϕiL*= logϕi(0)+ ω.logϕi(1) (5)
where ω is the acentric factor. The first term on the right hand side represents the fugacity coefficient of “simple fluids”. The second term is a correction accounting for departure of the properties of real fluids from those of “simple fluids”. (Grayson and Streed, 1963)
The quantity ϕi(0) depends only on reduced temperature
and reduced pressure. It was fitted with the following function by Chao and Seader (1961):
(6)
where Trand Prare the reduced temperature and pressure of
the component at hand. Coefficients for Equation (6) were determined by Grayson and Streed (1963) and they are presented in Table 1.
TABLE 1 Coefficients in Equation (6)
Coefficient Simple fluid Hydrogen
A0 2.05135 1.50709 A1 – 2.10899 2.74283 A2 0.0 – 0.02110 A3 – 0.19396 0.00011 A4 0.02282 0.0 A5 0.08852 0.008585 A6 0.0 0.0 A7 – 0.00872 0.0 A8 – 0.00353 0.0 A9 0.00203 0.0
Source: Grayson and Streed (1963), p. 177.
Special coefficients for Equation (6) are required for hydrogen since the typical application temperatures are far above the critical points of these two compounds. We know that the acentric factor, ω, for this component is not zero but in this work, the assumption of the original work is used and only logϕ(0) in Equation (5) is considered for hydrogen (Grayson and Streed, 1963).
The quantity ϕi(1)similarly depends only on reduced
temperature and reduced pressure and was fitted by Chao and Seader (1961): (7) log . . . . ( ) ϕi r r T T 1 4 23893 8 65808 1 22060 3 15 = − + ⋅ − − 2242 ⋅Tr3−0 025. ⋅
(
Pr−0 6.)
log . . . . ( ) ϕi r r r r A A T A T A T A T A A 0 0 1 2 3 2 4 3 5 6 = + + + + +(
+ TTr+A T7 r)
⋅ +Pr(
A +A Tr)
Pr − Pr 2 8 9 2 . . . log2.1.2 Activity Coefficient from Regular Solution Theory The liquid activity coefficient is calculated from the Hildebrand equation, assuming a “regular” liquid solution (no excess volume and no excess entropy). This equation is a liquid phase “molecular” model, proposed by Scatchard (1931) as a result of the work of Hildebrand (1916). It is essentially based on the concepts formulated by Van Laar (Scatchard and Hildebrand, 1934):
(8) where vi is the molar volume of component i, δi is its
solubility parameter and –δis the solubility parameter for the solution, which is calculated as follows (Walas, 1985):
(9) The quantity φiis the volume fraction, i.e. the ratio of the
molar volume of component i to the weighted molar volume of the mixture. The solubility parameter depends on the tem-perature but only its value at 25°C is usually taken as the difference between solubility parameters is almost independent of temperature (de Hemptinne et al., 2012). The quantities δi
and viare available in most databases for pure components at
25°C. The parameters used in this work are shown in Table 2. The solubility parameter can also be calculated from following relation (Riazi, 2005):
(10) where Δhivapis the molar heat of vaporization of component
i. As shown in Equation (10), the solubility parameter (δi) has
a physical meaning. Energy of vaporization is directly related to the energy required to overcome forces between molecules in the liquid phase and molar volume is proportional to the molecular size. Therefore, when two components have simi-lar values of δ their molecular size and forces are very similar yielding an ideal mixture.
The regular solution method is well adapted for non-polar components. It is very powerful in that it only uses physically meaningful pure component parameters. It can therefore be considered as predictive. The following limitations must be borne in mind however:
– it only predicts positive deviations from ideality (γi> 1),
i.e., only enthalpy deviation;
– it is not applicable to mixtures of polar components that generally show large deviations from ideality.
2.1.3 Fugacity Coefficient in a Vapour Phase
The Redlich and Kwong equation of state is employed for the calculation of the fugacity coefficient in the vapor mixture. This
coefficient is derived in terms of the compressibility factor Z following the standard procedure (Chao and Seader, 1961):
(11) Z3– Z2+ (A – B – B2) Z – AB = 0 (12) (13) (14) (15) (16) (17) where Triand Priare the reduced temperature and reduced
pressure of component i. B P T i ri ri =0 08664. ⋅ A P T i ri ri =0 42748. ⋅ 2 5. Aij= A Ai j B=
∑
( )
y Bi i A=∑
∑
(
y y Ai j ij)
ln ln ln ϕi V i i i Z B B Z B A B B B A A =(
−)
−(
−)
+ ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥⋅ 1 2 ⎛1++ ⎝ ⎜ B⎞⎠⎟ Z 2.2 Some ResultsThe Grayson Streed model is evaluated using experimental data on the solubility of hydrogen in pure hydrocarbons. The physical properties of each pure component are summarized in Table 2. The data sources for hydrogen solubility in vari-ous pure hydrocarbons are given in Table 3. These data are taken from: Grayson and Streed (1963) for hydrogen, n-heptane, n-decane and n-hexadecane. Yao et al. (1977-1978) reported the physical properties for 1-methyl naphthalene, Park et al. (1995) for eicosane, octacosane and hexatriacon-tane. Park et al. (1996) for phenanthrene and pyrene. The normal boiling temperatures (Tb) are taken from Riazi
(2005), it must be observed at this point that even though the trend of these properties with increasing molecular weight and within a given family, is monotonous, this is not the case for solubility parameters. A maximum is visible for eicosane. This trend is also visible in the data originating from the DIPPR (Design Institute for Physical PRoperties) database.
Since the gas solubility is essentially proportional to the pressure, it is often convenient to use a quantity that considers the ratio of pressure and solubility. The Henry constant of a gas is defined rigorously as follows:
(18) where Hiis the Henry constant for component i (solute). It
has the unit of pressure. Therefore, Hiis in fact the slope of
fiLversus xiat xi= 0. Note that the pressure is necessarily the
H f x i x i L i i = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ → lim 0 TABLE 2
Physical properties of pure components
Fluid name Formula MW Tb
(5) P
c Tc
ω v δ
(g/mol) (°C) (Pa) (K) (m3/kmol) (J/m3)0.5
Hydrogen(1) H 2 2.02 - 1 315 524 33.4 0 0.0310 6648 n-Heptane(1) C 7H16 100.2 98.4 2 735 849 540.2 0.3403 0.1475 15 300 n-Decane(1) C 10H22 142.3 174.2 2 096 013 618.9 0.4869 0.1960 15 793 n-Hexadecane(1) C 16H34 226.4 286.9 1 420 325 723.9 0.7078 0.2942 16 343 Eicosane(2) C 20H42 282.6 343.8 1 117 000 770.5 0.8738 0.3598 16 500 Octacosane(2) C 28H58 394.7 431.6 826 000 845.4 1.1073 0.5063 16 200 Hexatriacontane(2) C 36H74 506.9 497.1 682 000 901.1 1.2847 0.6484 16 200 1-Methylnaphthalene(3) C 11H10 142.2 244.7 3 252 533 772.2 0.3020 0.1399 20 046 Phenanthrene(4) C 14H10 178.2 339.9 3 300 000 873.2 0.5400 0.1580 20 000 Pyrene(4) C 16H10 202.2 392.8 2 600 000 938.2 0.8300 0.1584 19 670
MW: molecular weight. Tb: normal boiling temperature. Pc: critical pressure. Tc: critical temperature. ω: acentric factor. v: molar volume. δ: solubility parameter.
(1)Source: Grayson and Streed (1963), p. 175. (2)Source: Park et al. (1995), p. 243. (3)Source: Yao et al. (1977-1978), p. 300. (4)Source: Park et al. (1996), p. 72. (5)The normal boiling temperature (T
vapour pressure of the solvent (Psσ), because this is the
system pressure when the solute concentration reaches zero, as required by the definition. It is considered only as a function of temperature (Riazi, 2005).
Yet, in reality, it is often assumed that the fugacity can be written as a partial pressure (this is particularly true for hydrogen that behaves like an ideal gas up to significant pressures). Using the phase equilibrium relationship, the Henry constant can therefore be estimated from experimental data using:
(19)
where yiis the vapour mole fraction of the gas and xiits mole
fraction in the liquid phase. When such experimental values for Henry constant that are plotted as a function of molecular weight at given temperature, a decreasing trend is observed for n-alkanes (Fig. 1). Only the first component (n-heptane) does not follow the trend. This is probably the result of the fact that the hydrogen solubility range for these data is smaller, indicating that the Henry regime is not fully reached for all data. We may conclude from this observation that the uncertainty on the “experimental” Henry constants is rather large. This is why only the trends will be investigated rather than the actual values. Regressions will be performed on the solubility data rather than on the Henry constants.
Taking into account the definition of fiLshown on the right
hand side of Equation (3), the Henry constant can be re-written as follow:
(20) where γi∞ is the activity coefficient of component i (here
hydrogen) at infinite dilution, i.e., when xi→ 0 and fiL*is the
Hi i fi L =γ∞ * H y P x i i i =
pure component liquid fugacity of the solute i. A relationship between Henry constant (Hi) and the pure liquid fugacity
coefficient (ϕiL*) can be derived from Equation (20) and the
definition of this fugacity coefficient (ϕiL*):
(21) As mentioned earlier, the pressure is here necessarily the vapour pressure of the solvent (Psσ). Subscript i refers to the
gaseous component diluted in the solvent, in this work, from here onwards, we will use index 1 to refer to the solute (hydro-gen) and index 2 for the solvent. Equation (21) is a useful relationship between Henry’s law and the Grayson Streed model through the pure liquid fugacity coefficient (ϕ1L*).
ϕ γ σ σ i L i L s i s i f P H P * * = = ∞ TABLE 3
Source of experimental data on the solubility of H2in pure hydrocarbons
HC type Fluid name Nc Temperature Range pressure H2solub. Data source (°C) (bar) (mol%)
n-Alkane n-Heptane 7 150-200 3.7-40.5 0.0-5.50 Zernov et al. (1990)
n-Alkane n-Decane 10 71-310 1.5-255 0.0-50.1 Sebastian et al. (1980) & Park et al. (1995)
n-Alkane n-Hexadecane 16 189-391 20-254 3.11-51.9 Lin et al. (1980a)
n-Alkane Eicosane 20 50-300 10-129 1.13-12.9 Park et al. (1995) & Huang et al. (1988)
n-Alkane Octacosane 28 75-300 10-131 1.49-17.3 Park et al. (1995) & Huang et al. (1988)
n-Alkane Hexatriacontane 36 100-300 10-168 1.54-22.7 Park et al. (1995) & Huang et al. (1988) Aromatic 1-Methylnaphthalene 11 189-457 0.25-278 0.00-33.6 Yao et al. (1977-1978) & Lin et al. (1980b) Aromatic Phenanthrene 14 110-200 26-252 0.84-8.40 Park et al. (1996) & Malone and Kobayashi (1990) Aromatic Pyrene 16 159 52-197 1.59-5.75 Park et al. (1996)
Overall 7-36 50-457 0.25-278 0.00-51.9
Nc: carbon number. HC type: hydrocarbon type.
500 600 400
300
Exp. data for n -alkanes GS model AGS model 200 100 0 Henry constant of H 2 in n -alkanes, H1 (MPa) 130 50 90 110 70
Molecular weigth, MW (g/mol) Figure 1
On the other hand, rearranging Equation (21) makes it possible to understand how the Grayson Streed model calculates the Henry constant:
(22) The product of ϕ1L*P2σγ1∞must follow the same decreasing trend as that of the Henry constant seen in Figure 1. The vapour pressure decreases with increasing hydrocarbon molecular weight, as shown in Table 4, but the hydrogen fugacity coefficient (ϕ1L*) increases. As a consequence, the
result of ϕ1L*P
2σis almost constant, so the trend is almost entirely controlled by the activity coefficient at infinite dilution (γ1∞), i.e., γ1∞ is the driving element on the Henry constant. The product ϕ1L*P
2σγ1∞does not follow the correct trend using the Hildebrand model for activity coefficient (dotted line in Fig. 1): it rises continuously until eicosane that has the largest solubility parameter and shows a small decrease beyond.
3 THE FLORY AUGMENTED GS MODEL (AGS) 3.1 Justification
The non-ideal behaviour of a mixture (expressed in terms of excess Gibbs energy) is driven by two major contributions: enthalpic and entropic. This is a direct consequence of the Gibbs energy calculation as a sum of two terms (de Hemptinne et al., 2012):
GE= HE– TSE (23)
where GEis the excess Gibbs energy (J); HEis the excess
enthalpy (J) and SEis the excess entropy (J/K). As a result,
the activity coefficient itself can be written as the sum of an enthalpic (or “residual”) and entropic (or “combinatorial”) contribution:
lnγi= lnγires+ lnγicomb (24)
In the case of regular solutions (Eq. 8), the activity coefficient takes into account only an enthalpic (or “residual”)
H L P
1=ϕ1 2σγ1∞ *
contribution as cause of the non-ideal behaviour. However, when the sizes or shapes of molecules are different, an entropic contribution is also expected. In the case of mixtures of large hydrocarbons with hydrogen (very small molecule) the effect of different sizes may be expressed using the Flory model. The final model is simply a combination of the Hildebrand regular solution model (enthalpic) and the Flory model (entropic):
(25) where φiagain represents the volume fraction of component i
(see Eq. 9).
The first term on the right hand side of Equation (25) is a positive contribution due to differences in interaction energy; the term between brackets is a negative contribution due to molecular size differences. The combination of these terms makes it possible to represent either positive or negative deviation from Raoult’s law (Robinson and Chao, 1971).
The Flory contribution, as expressed in the second term of Equation (25) is fully predictive in the same way as the first term, which is the Hildebrand contribution. The activity coefficient model that uses the two terms is therefore a very interesting tool for evaluating the trends observed in the non-ideality of mixtures when both entropic and enthalpic effects are present.
In order to improve the required trend for the infinite dilution activity coefficient (Tab. 4), the addition of an entropic contribution will help: this contribution yields a value that is increasingly small as the solvent molar volume increases, as is shown in Table 5.
We can observe that the decreasing trend that is expected for the experimental Henry constant values is much better observed. This is why it is proposed to add a Flory-type entropic contribution as improvement on the calculation of the activity coefficient in the liquid phase.
However, Equation (22) now teaches us that if the value of the infinite dilution activity coefficient has changed, the reference value for the hydrogen liquid phase fugacity, ϕiL*,
lnγi δ δ lnφ φ i i i i i i v R T x x = ⋅
(
−)
⋅ + + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 1 TABLE 4Activity coefficient of H2at infinite dilution using the Hildebrand model at 423 K
Solvent Molecular weight, MW H2fugacity coefficient, Solv. vap. pressure, H2activity coefficient, ϕ1L*P2σγ1∞
should also be modified. This is done in this work by modifying the parameters of Equation (6): Table 1 is now replaced by Table 6.
3.2 Parameterization
The reference value for the hydrogen liquid phase fugacity, ϕ1L*, is adapted by modifying the parameters that are shown in Table 1. Only parameters A0and A1are adjusted because according to Equation (6) these parameters affect most strongly the temperature dependence of ϕ1L*. In addition, we
wanted to modify the original equation as little as possible. This modification is done by minimization of the sum of square errors between the predicted value by the AGS model (K1,calc)iand the experimental value (K1,exp)ion the solubility
of hydrogen in pure hydrocarbons. The objective function used to minimize and get the new parameters is as follow:
(26) where subscript 1 refers to hydrogen and subscript i refers to a condition of pressure and temperature (P,T). The experimental data selected for the parameterization was the same as shown in Table 2. The new coefficients to estimate the liquid fugacity coefficient of hydrogen are shown in Table 6.
3.3 Results
The correct trend of Henry constant with respect to molar mass is observed in Figure 1 (continuous line). Figure 2 presents an overall Absolute Average Deviation (AAD) of Henry constant for each binary system studied. It shows that the AGS model improves the prediction of Henry constant of hydrogen in heavy hydrocarbons, with an overall AAD equal to 11% for n-C16+ and heavy aromatics compounds. In the same way, the prediction of hydrogen solubility in heavy hydrocarbons is improved with the AGS model. A comparative evaluation is shown in Table 7.
FObj K cal i K i i n = ⎡⎣
(
)
−(
)
⎤⎦ =∑
ln 1, ln 1,exp 2 1 TABLE 5Flory contribution to activity coefficient of H2at infinite dilution in n-alkanes at 423 K
Solvent MW (g/mol) ϕ1L* P
2σ(Pa) γ1∞(Enthalpic) γ1∞(Entropic) γ1∞(global) ϕ1L*P2σγ1∞(Pa)
n-Heptane (C7H16) 100.2 105 374 830 1.934 0.4630 0.895 3.53 × 107 n-Decane (C10H22) 142.3 750 52 354 2.089 0.3670 0.767 3.01 × 107 n-Hexadecane (C16H34) 226.4 26 626 1 473 2.289 0.2578 0.590 2.31 × 107 n-Eicosane (C20H42) 282.6 289 269 135.6 2.352 0.2149 0.505 1.98 × 107 n-Octacosane (C28H58) 394.7 28 985 063 1.353 2.234 0.1565 0.350 1.37 × 107 n-Hexatriacontane (C36H74) 506.9 2 042 692 112 0.0192 2.234 0.1239 0.277 1.09 × 107
Note: ϕ1L* is here calculated using the parameters of Table 1.
AroC16 AroC14 AroC11 n-C36 n-C28 n-C20 n-C16 n-C10 n-C7 AAD (%) 80 0 60 70 50 40 20 30 10 Solvent GS model AGS model Figure 2
AAD of Henry constant of H2in binary mixtures of H2+
pure hydrocarbon.
TABLE 6
Coefficients for the AGS model in Equation (6) for Hydrogen
Considering heavy n-alkanes (n-C16+) and heavy aromatic compounds, the AAD for the AGS model is 16% vs 36% for the original GS model, indicating a 56% improvement, in the range of 50-457°C and 0.25-278 bar.
The deviations for the low molecular weight solvents are increased: the quality of the regular GS method for these sys-tems was already very good according to the available data. Yet, our focus was to improve the model for the heavy hydrocarbons, which is clearly a success. As a matter of fact, an improvement is visible for all aromatic solvents which were not included in the regression.
Figures 3 and 4 show a comparison between prediction solubility hydrogen with the AGS and the conventional GS models. The improvements with AGS are well visible. 4 APPLICATION TO PETROLEUM CUTS
The AGS model is evaluated on petroleum fractions: two Canadian heavy oils Light Virgin Gas Oil (LVGO) and Heavy Virgin Gas Oil (HVGO) which are typical feed stock for hydroprocessing, a Chinese Heavy Oil (GuDao Atmospheric Residuum, GDAR) and Athabasca Bitumen Vacuum Bottoms (ABVB). Relevant physical properties for these materials are given in Table 8. The normal boiling temperature is esti-mated from the vapour pressure curves shown by Cai et al. (2001). Note that these are in fact bubble temperatures of complex mixtures: they are not average boiling temperatures for the entire cut. Nevertheless, they provide an interesting value which is measured rather than calculated.
4.1 Petroleum Cut Characterization
A pseudo-component method based on the assumption that the petroleum fraction is a single, molecularly homogeneous pseudo-component is used in order to estimate the
characteristic properties (Pc, Tc, ω, v and δ) of the petroleum
fractions.
In this approach, the liquid phase is considered to be a mixture only of two components: the solute (dissolved gas,
i.e., hydrogen) and the solvent (petroleum fraction), which
are denoted again as components 1 and 2, respectively. It is known that many methods exist for determining these characteristic properties, as discussed for example by Riazi (2005). Each method yields potentially different characteristic parameters and therefore a more or less different end result. It would be a very extensive job to evaluate all methods for each characteristic property individually. Instead, we first propose a sensitivity analysis. This will allow us, in a second step, to focus our attention on the property which has the largest effect.
TABLE 8
Characterization of petroleum fractions
Property Unit LVGO HVGO GDAR ABVB
C wt% 85.0 84.4 85.4 84.3 H wt% 13.2 10.8 11.4 10.9 N wt% 0.4 1.5 0.8 0.8 S wt% 1.3 3.8 2.5 3.5 Density, ρ at 20°C g/cm3 0.892 0.973 0.922 1.05 Distillation range °C 184-454 274-595 350+ 525+ Normal boiling temperature °C 239.3 340.0 267.5 387.4 Mean molar mass g/mol 250 350 1 678 1 700 Aromatic carbon % 15.9 25.4 36.5 35.0 Inorganic solids ppm ~ 20 ~ 20
H/C ratio mol/mol 1.74 1.52 1.60 1.54
Source: Cai et al. (2001), p. 1 509.
TABLE 7
AAD on prediction of H2solubility in pure hydrocarbons
System Number of data Temperature Range pressure H2solubility (mol%)
% AAD
(°C) (bar) Grayson Streed AGS
a) b) c) d) 25 30 20 15 10 5 0 H2 Solubility , xH 2 (mol frac.) 0 0.02 0.04 0.06 0.08 0.10 0.12 Pressure, P (MPa) GS model Exp. data AGS model 25 30 20 15 10 5 0 H2 Solubility , xH 2 (mol frac.) 0 0.02 0.04 0.06 0.08 0.10 0.12 Pressure, P (MPa) GS model Exp. data AGS model 25 30 20 15 10 5 0 H2 Solubility , xH 2 (mol frac.) 0 0.12 0.14 Pressure, P (MPa) GS model Exp. data AGS model 25 30 20 15 10 5 0 H2 Solubility , xH 2 (mol frac.) 0 0.12 0.10 0.10 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0.14 Pressure, P (MPa) GS model Exp. data AGS model Figure 4
Solubility of hydrogen in phenanthrene at different temperatures. a) 398 K, b) 423 K, c) 448 K, d) 473 K.
12 14 10 8 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.05 0.10 0.15 0.20 Pressure, P (MPa) GS model Exp. data AGS model 12 14 10 8 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.05 0.10 0.15 0.20 Pressure, P (MPa)(MPa) GS model Exp. data AGS model 12 14 10 8 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.05 0.10 0.15 0.20 Pressure, P (MPa) GS model Exp. data AGS model 5 6 4 3 2 1 0 H2 Solubility , xH 2 (mol frac.) 0 0.05 0.10 0.15 Pressure, P (MPa)(MPa) a) b) c) d) GS model Exp. data AGS model Figure 3
4.2 Sensitivity Analysis
This analysis is performed using a base case, in this case H2 in HVGO and testing the effect of a 10% perturbation on each parameter on turn: solubility parameter (δ), critical pressure (Pc), critical temperature (Tc), acentric factor (ω),
molar volume (v) and molecular weight (MW) of the solvent. The characterization methods used for the sensitivity analysis are listed in Table 9 and a brief description of these methods is given in Appendix A, except for the solubility parameter which is detailed below.
An average on the relative deviation in the calculated hydrogen solubility is computed over all pressure and tempera-ture conditions given by Cai et al. (2001).
Table 10 shows the relative deviation on the prediction of hydrogen solubility as response of the perturbation in the parameters. This table shows that the solvent solubility parameter (δ), molar volume (v) and molecular weight (MW), are the most sensitive parameters to estimate the solubility of hydrogen in petroleum fractions. Critical temperature (Tc)
has also a non-negligible effect on the hydrogen solubility.
TABLE 9
Method to calculate physical properties of heavy oils
Property Method Source
Critical pressure Pc API Riazi (2005)
Critical temperature Tc API Riazi (2005) Acentric factor ω Korsten Riazi (2005) Molar volume at 25°C v Density and molecular weight Riazi (2005) Solubility parameter δ Definition Equation (10)
TABLE 10
Relative deviation of H2solubility due to parameters perturbation
Property Perturbation 10% – 10% Solubility parameter at 25°C δ2 – 23% + 24% Molar volume at 25°C V + 8.0% – 8.1% Molecular weight MW + 6.8% – 7.0% Critical temperature Tc + 1.5% – 6.1% Acentric factor ω + 0.2% – 0.2% Critical pressure Pc – 0.2% + 0.2%
4.3 Evaluation of the Characterization Methods for Petroleum Cuts
Looking at Table 8, it is visible that the lightest cut investi-gated is LVGO and the heaviest ABVB. Regarding the two other cuts (HVGO and GDAR), the data seem contradictory: density and normal boiling temperature of GDAR are lower,
but molar mass is significantly higher than that of HVGO. The aromatic carbon content is larger, yet the larger H/C ratio indicates it should be lower.
Yet, the resulting properties in Table 11 seem to indicate that ABVB is the heaviest (from the large Tcand acentric
factor), while GDAR is slightly heavier than LVGO. This can be readily explained by noting that the characterization method uses density and normal boiling point as input, rather than molar mass. Note that the solubility parameters according to method 1 (Eq. 10) are much too low compared to what is expected from hydrocarbons as shown in Table 3.
As a result of the sensitivity analysis, it appears that the solubility parameter is clearly the most important parameter. This is why three different methods have been tested for calculating this parameter, as shown below the horizontal line of Table 11.
TABLE 11
Physical properties calculated for petroleum fractions
Property Unit LVGO HVGO GDAR ABVB Critical pressure Pc MPa 2.40 2.50 2.62 3.12 Critical temperature Tc K 723.9 806.8 741.2 852.4 Acentric factor ω 0.431 0.916 0.644 1.234 Molar volume at 25°C v m3/kmol 0.281 0.361 1.826 1.622
Solubility parameter1at 25°Cδ (J/m3)0.5 14 211 16 808 6 284 9 229
Correction factor2 α 0.638 0.777 2.622 2.653
Solubility parameter3at 25°Cδ (J /m3)0.5 16 452 16 752 17 467 17 470
1Calculated by definition (Eq. 10, 27, 28), using Riedel and Watson method for ΔHvap.
2α correction factor modifies the solubility parameter of hydrogen (δ
H2) as follow: δH2, corrected= α.δH2. Riazi and Roomi (2007).
3Calculated by SCN method (Eq. 31), taken from Riazi and Vera (2005).
4.3.1 Solubility Parameter using the Definition
The definition of the solubility parameter is given by Equation (10). It is a function of enthalpy of vaporization at 25°C, which can be calculated in two steps: calculation at the normal boiling point, Δhbvap and subsequent correction to
25°C. One of most successful correlations for prediction of Δhbvapwas proposed by Riedel (Smith et al., 1996):
(27)
where R is the universal gas constant and Pcis the critical
pressure in bar. The unit of Δhbvapdepends on the unit of R
and Tc. Once Δhbvapis determined, the Watson relation can be
used to calculate Δhvapat the desired temperature (T) (Smith
4.3.2 Riazi Correction
We have seen that the activity coefficient in fact depends on the difference between solubility parameters (Eq. 25). Riazi and Roomi (2007) suggest to modify the gas solubility coefficient instead of that of the solvent. They state that the solubility parameter of the gas in a fictitious liquid phase (δ1) will depend on the molecular weight of this phase. This correction factor is introduced as:
δ1= αδ1(GS) (29)
where δ1is the corrected solubility parameter for hydrogen, α is the correction factor and δ1(GS)is the solubility parameter reported by Grayson and Streed (1963). The correction factor depends on the type of gas and solvent (i.e., a petroleum frac-tion or a coal liquid) but is independent of temperature. For paraffin rich petroleum fractions following relationship is proposed by Riazi and Roomi (2007):
α = 0.29 + 0.00139MW (30)
This relationship has been developed for paraffinic crudes only because of the lack of sufficient data on naphthenic or aromatic crudes (Riazi and Roomi, 2007).
The petroleum fractions investigated in this work have a relatively low aromatic content (< 37%) making it possible to use this correlation.
The solubility parameter of the solvent (δ2) in this case is calculated using the definition (Eq. 10).
4.3.3 Single Carbon Number (SCN) Approach
The third correlation is based on Riazi and Al-Sahhaf (1996), where following relation is proposed to calculate δ2from the molecular weight of the liquid fraction:
δ2= 17.5913 – exp(3.0076 – 0.549097MW0.3) (31) where δ2 is expressed in (J/cm3)0.5. This correlation can be used within the molecular weight range of 80-700 (~C6-C50). This value of δ2is based on estimated values for properties of Single Carbon Number (SCN) groups (Riazi and Vera, 2005). Table 11 shows the results of the physical properties including the three options for solubility parameter calculation. It is clear that the solubility parameter value is very different for the three calculation modes. The results obtained from the SCN (Single Carbon Number) method are most reasonable, since the values of the solubility parameter as largest for the heaviest cut, ABVB.
The calculations from the definition are least reasonable because the values found are very small compared to those of hydrocarbons of similar molecular weight (see for example
Tab. 3).
4.4 Application to Coal Liquids
Additionally, AGS model is evaluated in two narrow boiling distillate cuts from the Exxon Donor Solvent (EDS) and three narrow boiling distillate cuts from the Solvent Refined Coal II (SRC-II) process. Relevant physical properties for these materials are given in Table 12.
TABLE 12 Characterization of coal liquids
Property Unit CLPP A5 CLPP A5 SRC-II SRC-II SRC-II
204-232°C 260-315°C No. 5 No. 9 No. 12
s.g. at 15°C 0.9320 0.9844 0.9826 1.0306 1.0910 Distillation curve (°C) 5% % 191.1 251.9 233.3 302.2 358.9 10% 197.9 258.7 235.6 303.3 360.0 30% 204.3 272.9 243.3 307.8 362.2 50% ppm 209.7 285.3 251.1 314.4 363.3 70% mol/mol 219.3 295.2 261.1 322.2 367.8 90% 230.4 313.2 280.0 336.7 382.2 95% 233.9 319.7 292.2 348.9 392.2
Mean molar mass g/mol 154.34 182.30 182 212 252
Total aromatics wt% 71.78 73.09
TABLE 14
AAD on prediction of H2Solubility in petroleum fraction
Petroleum
No. of data Temp. Range H2solub. δ2
by definition α factor on δ1 SCN approach
fraction (°C) pressure (bar) (mol%) GS AGS GS AGS GS AGS
LVGO 38 80-380 11.7-122.7 1.9-23.7 29% 45% 13% 13% 11% 14% HVGO 43 80-380 6.3-118.8 0.7-42.3 19% 30% 35% 15% 18% 31% GDAR 33 80-380 6.7-106.0 5.1-60.3 51% 156% 82% 15% 80% 14% ABVB 29 130-380 7.3-121.2 2.6-70.0 42% 166% 64% 64% 74% 24% CLPP(204-232°C) 9 190-270 49.6-258.9 0.04-0.22 76% 34% CLPP(260-315°C) 8 190-270 50.0-254.4 0.03-0.19 96% 62% SRC-II No. 5 8 190-270 50.7-254.0 0.04-0.20 82% 50% SRC-II No. 9 8 190-270 48.4-253.0 0.03-0.19 95% 74% SRC-II No. 12 5 270 51.7-252.7 0.04-0.16 119% 117% Overall 181 80-380 6.3-258.9 0.03-70.0 35% 73% 48% 23% 55% 30%
GS: Grayson Streed model.
AGS: Augmented Grayson Streed model.
TABLE 13
Physical properties calculated for coal liquids
Property Unit CLPP A5 CLPP A5 SRC-II SRC-II SRC-II
204-232°C 260-315°C No. 5 No. 9 No. 12
Critical pressure Pc MPa 2.94 2.50 2.86 2.56 2.54
Critical temperature Tc K 710.8 794.1 758.9 831.7 897.8
Acentric factor ω 0.364 0.448 0.390 0.455 0.486
Molar volume at 25°C v m3/kmol 0.167 0.186 0.186 0.207 0.232
Solubility parameter1at 25°C δ (J/m3)0.5 15 913 16 114 16 112 16 283 16 460
The results of physical properties calculated for the coal liquids are shown in Table 13, in this case the solubility parameter was calculated by SCN method, according to Equation (32).
4.5 Calculation Results and Discussion
Both the GS and AGS models have been run with these three different input data for petroleum fractions and Table 14 shows a summary of the calculations of hydrogen solubility in petroleum fractions, as well as hydrogen solubility in coal liquids. As can be seen, the hydrogen solubility prediction in petroleum fractions is very sensitive to the values of the physical properties. However, the best predictions are obtained with the SCN solubility parameter approach. Except when using the solubility parameter calculated “from the definition”, the predictions of hydrogen solubility are improved compared with the GS model.
For the SCN prediction method, which we recommend here, the Absolute Average Deviation (AAD) is 30% for AGS vs 55% the GS model, including the hydrogen solubility in coal liquids.
Figure 5 shows the prediction of hydrogen solubility in petroleum cuts at given temperatures and Figure 6 shows the Henry constant as a function of temperature predicted by GS and AGS for the petroleum fractions, using the SCN approach to the solubility parameter.
8 10 12 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.10 0.05 0.15 0.20 0.25 Pressure, P (MPa) GS model Exp. data AGS model 12 10 8 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.10 0.05 0.20 0.15 0.35 0.30 0.25 0.40 Pressure, P (MPa) GS model Exp. data AGS model 12 10 8 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.3 0.2 0.1 0.4 0.5 0.6 Pressure, P (MPa) GS model Exp. data AGS model 10 12 8 6 4 2 0 H2 Solubility , xH 2 (mol frac.) 0 0.2 0.1 0.4 0.3 0.5 0.6 Pressure, P (MPa) a) b) c) d) GS model Exp. data AGS model Figure 5
Solubility of hydrogen on heavy petroleum cuts. a) in LVGO at 603 K, b) in HVGO a 653 K, c) in ABVB at 523 K, d) in GDAR at 523 K.
600 700 500 400 300 Henry constant, H (MPa) 0 100 75 50 25 125 150 175 Temperature, T (K) GS model Exp. data AGS model 700 600 500 400 300 Henry constant, H (MPa) 0 100 50 150 200 250 Temperature, T (K) GS model Exp. data AGS model a) b) 600 700 500 400 300 Henry constant, H (MPa) 0 25 50 100 75 125 150 Temperature, T (K) GS model Exp. data AGS model 700 600 500 400 300 Henry constant, H (MPa) 0 125 100 75 50 25 150 175 200 Temperature, T (K) GS model Exp. data AGS model c) d) Figure 6
In summary, AGS is capable of predicting solubility of hydrogen in various petroleum cuts. However, the main advantage of this method is that in addition to simplicity and availability of input parameters it does not need binary interaction parameters: the approach using the solubility parameter is purely predictive.
CONCLUSION
The Grayson Streed model underestimates the solubility of hydrogen in heavy pure n-alkanes and overestimates the solubility of hydrogen in pure aromatic compounds.
In order to improve this behaviour, a simple methodology has been proposed, leading to a new model that reflects more correctly the effect of the entropic contribution when small gas molecules are dissolved in a heavy hydrocarbon solvent. This model, called “Augmented Grayson Street” is based on the addition of an entropic Flory term in the activity coefficient calculation.
New reference values for the hydrogen liquid phase fugacity coefficient, ϕ1L*, have been obtained as a result of a change in the infinite dilution activity coefficients for the AGS model.
The AGS model improves the predictions for solubility of hydrogen in heavy n-alkanes components (C16+) and also improves the solubility predictions in aromatics. Although a degradation is observed for low molecular weight alkane solvents, the overall AAD for AGS model is 16% vs 36% for GS model, indicating a 56% improvement, in the range of 50-457°C and 0.25-278 bar.
The main difficulty to use the model for hydrogen solubility calculations in petroleum fractions is to find adequate methods to estimate the physical properties of pseudo-component. The model is most sensitive to a correct representation of the solubility parameter for which several prediction methods have been tried. The SCN method (Riazi and Vera, 2005) was found best.
The predictions of hydrogen solubility in petroleum fractions and in coal liquids were improved compared with the Grayson Streed model, with AAD = 30% for AGS model compared to 55% for Grayson Streed model, in the range of 80-380°C and 6.3-258.9 bar.
It should be stressed that other approaches, based on homogeneous equation of state models, may lead to more accurate results. The goal of the analysis proposed in this work is to suggest that for heavy components the sole Hildebrand activity coefficient model is not sufficient.
ACKNOWLEDGMENTS
This work was performed as a master thesis in collaboration between IFP-School and PDVSA. Special thanks to Daniel Dumas who is responsible for this collaboration.
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Final manuscript received in August 2012 Published online in April 2013
Copyright © 2013 IFP Energies nouvelles
APPENDIX A
A.1 Prediction of Critical Pressure and Critical Temperature
Simplified equations to calculate Pcand Tcof petroleum fractions in the range of C5-C20 are given by following equations (Riazi, 2005):
(A.1)
(A.2) where Pcis the critical pressure (bar); Tbis the normal boiling temperature (K); SG is the specific gravity at 15.5°C (obtained
from density) and Tcis the critical temperature (K).
These equations are recommended only for hydrocarbons in the molecular weight range of 70-300. They were adopted by the API and have been used in many industrial computer softwares under the API method.
For heavy hydrocarbons (>C20) the following equations are recommended (Riazi, 2005):
(A.3) (A.4)
A.2 Prediction of Acentric Factor
The acentric factor, ω, is a measure of the shape of the vapour pressure curve. Values of the acentric factor can be obtained using Tc, Pcand vapour pressure. Most recently Korsten (Riazi, 2005) modified the Clapeyron equation for vapour pressure of
hydrocarbon systems and derived following equation:
(A.5) where ω is the acentric factor, Tbris the reduced normal boiling temperature (K) and Pcis the critical pressure (bar).
A.3 Prediction of Molar Volume
Molar volume at 25°C is calculated from density at the same temperature and molecular weight. If density is given at another temperature, following equation may be used to obtain density at any temperature once a value of density is known (Riazi, 2005):
(A.6) where ρ0is the density (g/cm3) at T
0(for example 298.15 K) and ρTis the density (g/cm3) at given temperature T.
A.4 Prediction of Solubility Parameter
Three methods to calculate the solubility parameter are considered as is shown in Table 11. They are described in the text.
